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%I #13 Feb 24 2015 10:46:03
%S 1,1,263130836933693530167218012160000000
%N Product_{k=0..n} (k^5)!.
%C The next term a(3) has 512 digits.
%C In general (for m>1), product_{k=0..n} (k^m)! ~ c(m) * (2*Pi)^(n/2) * n^(m*(1/4 + n/2 + B(m+1)/(m+1) + (sum_{j=1..n} j^m) )) * exp(-m*n/2 - m*n^(m+1)/(m+1)^2 - (sum_{j=1..n} j^m) + m * (sum_{j=1..m-1} 1/(j+1) * B(j+1) * binomial(m, j) * n^(m-j) * (sum_{i=0..j-1} 1/(m-i)) )), where c(m) is a constant and B(n) is the Bernoulli number A027641(n)/A027642(n).
%H Vaclav Kotesovec, <a href="/A255360/b255360.txt">Table of n, a(n) for n = 0..3</a>
%F a(n) ~ c * n^(80/63 + 5*n/2 - 5*n^2/12 + 25*n^4/12 + 5*n^5/2 + (5*n^6)/6) * (2*Pi)^(n/2) / exp(5*n/2 + 35*n^2/144 + n^5/2 + 11*n^6/36), where c = A255439 = 11.354954749729782312106... .
%t Table[Product[(k^5)!, {k, 0, n}], {n, 0, 4}]
%Y Cf. A000178, A255322, A255358, A255359.
%Y Cf. A002109, A051675, A255321, A255323, A255344.
%Y Cf. A000142, A255268, A255269, A255439.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Feb 21 2015
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